In a large mixing bowl, combine 2 cups of flour and yeast.
In a small saucepan, heat milk, butter, sugar, and salt until warm and butter is almost melted.
Add milk mixture to flour mixture along with eggs. Beat with an electric mixer on low speed for 30 seconds, then on high speed for 3 minutes.
Stir in remaining flour. Knead dough on a lightly floured surface until smooth and elastic.
Place dough in a greased bowl, cover, and let rise until doubled in size, about 1 hour.
Roll dough into a 12x8 inch rectangle. Spread softened butter over dough. Combine brown sugar and cinnamon, sprinkle over butter.
Roll up dough starting from the long side. Cut into 12 slices. Place in a greased baking dish. Cover and let rise until nearly doubled, about 30 minutes.
Bake in a 375°F (190°C) oven for 25 minutes or until golden brown.
For icing, combine powdered sugar, milk, and vanilla. Drizzle over warm rolls.